Publication 001 · Research Article
The Mirror Calculus: A Presence-Only Mathematical Language
A mathematical notation in which every mark is kept under left–right reflection and every numeral is formed by strokes [the numeral zero stands outside the inventory], carried from its grammar through geometry, algebra, analysis, and analytic number theory. Formation precedes assertion. An object-language inscription forms only through a positive witness, trace, relation, transformation, roster, pairing, certificate, or explicit sponsorship event; non-presentation forms no substitute term.
Numerical zero, empty collections, null returns, default falsity, vacuous judgments, and negative records inferred from failed search are analysed as surrogate-zero devices: they make non-presentation participate as an object, value, or verdict.
The controlling chain
The Riemann question is resolved at its own registers, each verdict carrying its sponsor on its face. The native axis sentence is a sentence of the presence grammar; the literal zero-locus sentence belongs to the classical register.
RHC(0) ∉ Sent(GP) · RHM ∈ Sent(GP)
Over the declared base, class-wide Admission and its counter-inscription are each witnessed by a finite model in exact rationals: a relative independence theorem, with both sides exhibited. The adopted theory P† = P + Admζ derives the native axis sentence — the resolution, a native proof:
the verdictP† ⊢ RHM
Under the named analytic interface the adopted law is equivalent to the classical hypothesis, proved from the interface definition, and the classical statement holds there with its sponsor carried on the face of the verdict. Why the sign was the obstacle, what the law is on the prime side, and the resolution stated at full strength with its trace are in Deprogramming Zero, the companion paper; the programme in one document is the Summary. The sponsor is the terminal constructor: by the Selection Jump a coherent family forms by a presented family, a generator with a presented uniform keeping law, or an explicit sponsor; the Emmerson–Buchanan barrier hierarchy forces the second into ladders; the third forms the seal. The verdict is sealed, sponsored, and armed — a presented off-axis record fires the clash channel — and that is the final form a verdict on this question takes.
I(Admζ) ⟺ RHC
An explicit polynomial witness carries every reflection and functional-equation symmetry the tent group imposes together with an off-axis quartet, and so measures exactly what Admζ supplies beyond symmetry.
Start here
- The hypothesis, stated four ways — three classical equivalents, and one native to a grammar in which every numeral is formed by strokes [the numeral zero stands outside the inventory].
- Why the content lies beyond symmetry — an explicit witness carrying every reflection symmetry, with roots off the line. Shown live.
- Glossary — Mirror Calculus, surrogate zero, presence-only formation, and the rest of the vocabulary.
- Open problems — the seven the volume posts, stated as mathematics, each with its own address; the Seven puts a lock and a prize on each.
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Companion: Future Directions — the flat page is saturated; new constructors for the folded carriers, with the machine check animated. The corner’s grammar forces Vieta.
116 numbered results · 305 engraved plates · machine-readable record in JATS 1.3 · MSC 2020: 03A05, 03B22, 03F65, 11M06, 11M26, 03B35
Applied Work · Free to play
Counterfoil
Counterfoil implements THE WINDOW (Contra-Zero Certificates): the same presence-only law as the mathematics, made playable. Two seats, forty-three certificates, and thirty strokes that move between a shared plenum and each player's hoard but are never made or spent out of existence.
What makes it unusual
- Every turn is an act. Every turn is exactly one action; if every action is closed to you, you are compelled to advance the clock — even when a card forbids it.
- Everything is open. Hands, chains, hoards, and captures are open to inspection at all times. Every claim on the table is enacted or standing open — each in its own name.
- Cost is weight, carried on the face. A card is built free. The number in the ring is what the card is worth standing, and what it costs you when someone takes it.
- Three columns, each counted under its own name. Substance, Form, and Holdings are compared one against one because they measure incommensurable things. A split table is a Balance — the table's third verdict, counted under its own name.
Cards are serial-numbered prints from fixed, published runs; the census is public and anyone may audit it. Finish is finish; play is play.
Below the ring
Beyond the tables lies the Descent, in design: six stages named for the adopted descent laws, each held by an adversary that is a surrogate-zero device given a body and a move set. Each imposes a distortion — a rule that breaks the law of the table — and beating it refuses the distortion. Someone who finishes has enacted the argument of the volume ahead of being taught a word of it.
Below the Descent stand the Seven: seven locks on the volume's seven open problems. Each holds a prize from the treasury and a certificate printed once; each sandbox holds the checker in full; the answer stays with you. Three release on a machine check, four on a read under the review terms. Entry is three hundred marks, the five-dollar tier.
Play in your browser The rules The forty-three certificates The Seven The rulebook
Public audits: card ledger · currency supply